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A Mélange of Diameter Helly-Type Theorems
SIAM Journal on Discrete Mathematics ( IF 0.9 ) Pub Date : 2021-07-19 , DOI: 10.1137/20m1365119
Travis Dillon , Pablo Soberón

SIAM Journal on Discrete Mathematics, Volume 35, Issue 3, Page 1615-1627, January 2021.
A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a long-standing conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of “containing $k$ colinear integer points.”


中文翻译:

直径螺旋型定理的混合体

SIAM Journal on Discrete Mathematics,第 35 卷,第 3 期,第 1615-1627 页,2021 年 1 月
。直径的 Helly 型定理提供了 $\mathbb{R} 中有限凸集族的交集的直径的界限^d$ 给出了所有足够小的亚族的交集直径的一些信息。我们证明了 Bárány、Katchalski 和 Pach 长期存在的猜想的分数和彩色版本。我们还表明,当且仅当其单位球是多面体时,Minkowski 范数承认直径的精确 Helly 型定理,并且证明了一个彩色版本。最后,我们证明了“包含 $k$ 共线整数点”的性质的 Helly 型定理。
更新日期:2021-07-19
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